Difference of Hilbert series of homogeneous monoid algebras and their normalizations

Let Q be an affine monoid and k[Q] the associated monoid k-algebra, where we let k be a field.
In this talk, in the case where k[Q] is standard graded, a difference of the Hilbert series of k[Q] and its normalization is discussed. More precisely, we prove that if k[Q] satisfies Serre’s condition (S2), then the degree of the h-polynomial of k[Q] is always greater than or
equal to that of its normalization.
Moreover, we also show counterexamples of this statement if we drop the assumption (S2).